1. COMPARING MODELS A cup of water at an initial temperature of is placed in a room at a

constant temperature of The temperature of the water is measured every 5 minutes

during a half-hour period. The results are recorded as ordered pairs of the

form where is the time (in minutes) and is the temperature (in degrees

Celsius).

(a) The graph of the model for the

data should be asymptotic with the graph of the temperature of the

room. Subtract the room

temperature from each of the temperatures in the ordered pairs. Use a graphing utility

to plot the data points and

(b) An exponential model

1. COMPARING MODELS A cup of water at an initial temperature of is placed in a room at a

constant temperature of The temperature of the water is measured every 5 minutes

during a half-hour period. The results are recorded as ordered pairs of the

form where is the time (in minutes) and is the temperature (in degrees

Celsius).

(a) The graph of the model for the

data should be asymptotic with the graph of the temperature of the

room. Subtract the room

temperature from each of the temperatures in the ordered pairs. Use a graphing utility

to plot the data points and

(b) An exponential model for the

data is given by Solve for and graph the model. Compare the result with the

plot of the original data.

(c) Take the natural logarithms of

the revised temperatures. Use a graphing utility to plot the points and observe

that the points appear to be linear. Use the regression feature of the

graphing utility to fit a line to these data. This resulting line has the form

Solve for and verify that the result is equivalent to the model in part (b).

(d) Fit a rational model to the

data. Take the reciprocals of the -coordinates of the revised data points to

generate the points Use a graphing utility to graph these points and observe

that they appear to be linear. Use the regression feature of a graphing

utility to fit a line to these data. The resulting line has the form Solve for

and use a graphing utility to graph the rational function and the original data

points.

(e) Why did taking the logarithms

of the temperatures lead to a linear scatter plot? Why did taking the

reciprocals of the temperatures lead to a linear scatter plot?

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